Question

a yo-yo is made of two uniform disks, each of mass m and radius r, which are glued to a smaller central axle of negligible mass and radius 0.5r (see figure). a string is wrapped tightly around the axle. the yo-yo is then released from rest and allowed to drop downwards, as the string unwinds without slipping from the central axle.

Answer 1

To determine the acceleration of the yo-yo as it falls and unwinds, consider the forces acting on it. The torque from the tension in the string provides the necessary force for the angular acceleration. By setting the torque equal to the moment of inertia times the angular acceleration, the angular acceleration can be calculated. Using the relationship between linear and angular acceleration, the linear acceleration can then be found.

Step-by-step explanation:

In order to determine the acceleration of the yo-yo, we can consider the forces acting on it. There are two forces at play: the weight (mg) acting downward and the tension in the string acting upward. Since the yo-yo is falling without slipping, the tension in the string provides the necessary force for the angular acceleration. The equation for the angular acceleration is given by α = Στ/I, where α is the angular acceleration, Στ is the torque, and I is the moment of inertia.

Since the yo-yo is a solid cylinder, the moment of inertia is given by I = (1/2)mr². The torque can be calculated by considering the perpendicular distance from the axis of rotation to the point where the force is applied. In this case, the radius of the axle is 0.5r, so the torque is given by Στ = F x (0.5r).

Setting the torque equal to Iα, we can solve for the angular acceleration. Once we know the angular acceleration, we can use the relation between linear and angular acceleration to find the linear acceleration, a = rα.